I have not found a proof of precisely this identity, but it follows quickly from the equations in <A HREF="http://www.ams.org/journals/proc/2001-129-03/S0002-9939-00-05821-4/S0002-9939-00-05821-4.pdf">
Orthogonal polynomials on the unit circle associated with the Laguerre polynomials
</A>. The Fourier cosine and sine transforms of $\phi_m$ are equal to the real and imaginary part of $(1-z)z^m$, with $z=(2k-i)/(2k+i)$.